# Finitely Generated Abelian Groups

1. Mar 4, 2012

### Combinatorics

1. The problem statement, all variables and given/known data

Let $$p$$ be prime and let $$b_1 ,...,b_k$$ be non-negative integers. Show that if :
$$G \simeq (\mathbb{Z} / p )^{b_1} \oplus ... \oplus (\mathbb{Z} / p^k ) ^ {b_k}$$
then the integers $$b_i$$ are uniquely determined by G . (Hint: consider the kernel of the homomorphism $$f_i :G \to G$$ that is multiplication by $$p^i$$ . Show that $$f_1 , f_2$$ determine $$b_1$$.
Proceed similarly )

2. Relevant equations
The question, together with the theorem it should help me prove is attached (this question should help me prove the uniqueness part of the theorem)

3. The attempt at a solution
I've tried using the hint, and considering the homomorphism $$f_1 : G \to G$$ that is defined by $$f_1 (x ) = px$$ . Its kernel is $$ker f_1 = (\mathbb{Z}_p ) ^{b_1} \bigoplus (\mathbb{Z}_p) ^ {b_2} ...$$.
But how does it help me? Am I right in my calculation of the kernel?

Hope someone will be able to help me

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2. Mar 9, 2012

### morphism

Re: Finitely-Generated-Abelian-Groups

The hint is essentially leading you to consider a proof by induction on k. Notice that $$\ker f_1 = (\mathbb Z/p)^{b_1} \oplus (\mathbb Z/p)^{b_2} \oplus (\mathbb Z/p^2)^{b_3} \oplus \cdots = (\mathbb Z/p)^{b_1+b_2} \oplus (\mathbb Z/p^2)^{b_3} \oplus \cdots.$$

3. Mar 9, 2012

### Combinatorics

Re: Finitely-Generated-Abelian-Groups

Can you please explain how did you get that this is the kernel?

Thanks a lot !

4. Mar 9, 2012

### morphism

Re: Finitely-Generated-Abelian-Groups

Well, if px=0 mod p^i, then p^i divides px, hence p^{i-1} divides x, i.e. x=0 mod p^i.